Abstract
In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.
Highlights
In recent years, impulse theory has been significantly developed, especially in the cases of impulsive differential equations or differential inclusions with fixed moments; see the monographs of Lakshmikantham et al [1], Samoilenko and Perestyuk [2] and Perestyuk et al [3] and the references therein
Some extensions to impulsive differential equations with variable times have been done by Bajo and Liz [4] and Frigon and O’Regan [5,6], and in the multivalued case, for instance, by
We will be in the presence of "pulse accumulation" whenever a solution has an infinite number of pulses which accumulate to a finite time t∗
Summary
Impulse theory has been significantly developed, especially in the cases of impulsive differential equations or differential inclusions with fixed moments; see the monographs of Lakshmikantham et al [1], Samoilenko and Perestyuk [2] and Perestyuk et al [3] and the references therein. In this paper we consider a class of initial value problems (IVPs) for differential equations with impulses at variable times on [ a, b], allowing pulse accumulation:. Note that usual IVPs should be treated as impulsive problems with negligible jumps In this case the space C ([ a, b]) or C1 ([ a, b]) are considered, and they are subspaces of G ([ a, b]). IVPs with impulses at finite and fixed times have been studied in the subspace PC ([ a, b], t1 , t2 , ..., tk ) of the space PC ([ a, b]) of piecewise continuous functions, so that the space of solutions depends on times of jumps. Requires that the sum of jumps (left and right) is finite and this condition implies that any solution is continuable to the point b. We still get a global solution for the impulsive problem
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